It follows from Lemmas REF  and REF  that either assertions (1.b) or (1.c) hold, or \(Y\)  is isomorphic to one of the
groups listed in Examples 2.2 and 2.4 of [1]}.
Assume that \(e=n\) . If \(Y\)  is isomorphic to one of the groups listed as in Example 2.2 of [1]}, then \(Y\)  is the stabilizer of a subspace or a quotient of a subspace \(V_{n}(q)\)  of dimension \(m\geqslant e\) , and hence this case cannot occur for \(e=n\) .
Then \(Y\)  is listed in Example 2.4 of [1]} and hence \(Y\leqslant GL_{n/c}(q^{c})\cdot Z_{c}\)  for some divisor \(c\)  of \(n\)  with \(c>1\) . Clearly,
\(\Phi _{nf}^{\ast }(p)\)  divides the order of \(GL_{n/c}(q^{c})\cdot Z_{c}\) .
Then \(\Phi _{nf}^{\ast }(p)\)  divides the order of \(SL_{n/c}(q^{c})\) , as \(\Phi _{nf}^{\ast }(p)\equiv 1 \pmod {nf}\)  by [4]}, Proposition
5.2.15.(i). Also \(\Phi _{nf}^{\ast }(p)\)  is coprime to \(\left[SL_{n}(q):SL_{n/c}(q^{c})\right] \) , and hence \(\Phi _{nf}^{\ast }(p)^{1/2}\) 
divides the order of \(Y\cap SL_{n/c}(q^{c})\) , since \(Y\leqslant GL_{n/c}(q^{c})\cdot Z_{c}\) .
Set \(f^{\prime }=cf\) . Clearly, \(\Phi _{nf}^{\ast }(p)=\Phi _{\frac{n}{c}f^{\prime }}^{\ast }(p)\)  and \(\Phi _{\frac{n}{c}f^{\prime }}^{\ast }(p)^{1/2} \)  divides \(\left|Y\cap SL_{n/c}(q^{c})\right|\)  and \(\left[ SL_{n/c}(q^{c}):Y\cap SL_{n/c}(q^{c})\right] \) . Hence, we may apply
Lemmas REF  and REF  with \(Y\cap SL_{n/c}(q^{c})\)  and \(SL_{n/c}(q^{c})\)  in the role \(Y\)  and \(SL_{n}(q)\)  respectively, thus obtaining \(Y\cap SL_{n/c}(q^{c})\leqslant GL_{n/c^{\prime }}(q^{c^{\prime }})\cdot Z_{c^{\prime }}\)  with \(c\mid c^{\prime }\) . Indeed, no cases arise from Lemma REF , since \(c>1\) . We may iterate the previous argument and eventually obtain that \(Y\leqslant GL_{1}(q^{n})\cdot Z_{n}\)  and \(p\mid n\) , as \(p\mid \left|Y\right|\) , which is (1.a).
Assume that \(e=n-1\) . If \(Y\)  is a group listed in Example 2.2 of [1]},
then \(Y\)  is the stabilizer of a subspace or a quotient of a subspace \(V_{n}(q)\)  of dimension \(n-1\)  and we obtain (2).
Assume that \(Y\)  is listed in Example 2.4 of [1]}. Let \(u\)  is prime
divisor of \(\Phi _{(n-1)f}^{\ast }(p)\) , then \(u=n\)  and \(Y\leqslant GL_{1}(q^{n-1})\cdot Z_{n-1}\)  by Example 2.4(a) of [1]}. Since \(Y \)  contains a normal Sylow \(u\) -subgroup \(U\)  which stabilizes a unique
hyperplane of \(V_{n}(q)\) , we obtain (2).
Assume that \(e=n-2\) . If \(Y\)  is one of the groups listed in Example 2.2 of [1]}, then \(Y\)  is the stabilizer of a subspace or a quotient of a subspace
\(Q\)  of \(V_{n}(q)\)  of dimension \(m=n-1\)  or \(n-2\) . In the latter case \(Y\)  lies in a subgroup of type \(P_{2}\)  of \(SL_{n}(q)\) , and hence (3) holds.
Assume that \(m=n-1\) . Suppose that \(\dim Q=1\) . Then \(Y\leqslant [q^{n-1}]:GL_{n-1}(q)\) , with \([q^{n-1}]\)  fixing \(Q\)  pointwise, by [4]}, Proposition 4.1.17.(II). Set \(W=Y/(Y\cap [q^{n-1}])\)  and \(n^{\prime }=n-1\) . Hence \(e=n-2=n^{\prime }-1\)  and \(W\)  acts on \(V_{n}(q)/Q\)  inducing a subgroup of \(GL_{n^{\prime }}(q) \) . Moreover, arguing as in the \(e=n-1\)  case, we see that \(W\)  is the
stabilizer of a subspace or a quotient of a subspace \(V_{n}(q)/Q\)  of
dimension \(n^{\prime }-1\) . Hence \(Y\)  lies in a parabolic subgroup of type \(P_{2}\)  or \(P_{1,n-1}\)  of \(SL_{n}(q)\) , which is (3). The same conclusions hold for \(\dim Q=n-1\) .
Finally, assume that \(Y\)  is one of the groups in Example 2.4 of [1]}. Then \(n\) 
is even and \(Y\leqslant GL_{n/2}(q^{2})\cdot Z_{2}\) . If \(f^{\prime }=2f\) , then \(\Phi _{\left(n/2-1\right) f^{\prime }}^{\ast }(p)^{1/2}\mid \left( \left|Y\right|,\left[ SL_{n/2}(q^{2}):Y\right] \right) \)  and \(p\mid \left|Y\right|\) . Arguing as in the \(e=n-1\)  case, we get that \(Y\) 
fixes subspace or a quotient of a subspace \(V_{n/2}(q^{2})\)  of dimension \(n/2-1\) . Then \(Y\)  is lies in a parabolic subgroup of type \(P_{2}\)  of \(SL_{n}(q)\) , and we obtain (3).
